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In: Statistics and Probability

The usable lifetime of a particular electronic component is known to follow an exponential distribution with...

The usable lifetime of a particular electronic component is known to follow an exponential distribution with a mean of 6.1 years. Let X = the usable lifetime of a randomly selected component. (a) The proportion of these components that have a usable lifetime between 5.9 and 8.1 years is __. (b) The probability that a randomly selected component will have a usable life more than 7.5 years is __. (c) The variance of X is __.

Solutions

Expert Solution

Given that the usable lifetime of a particular electronic component is known to follow an exponential distribution with a mean of 6.1 years.

Let,

X = the usable lifetime of a randomly selected component.

Before we go on to solve the problems let us know a bit about Exponential Distribution.

Exponential Distribution

A positive continuous random variable X is said to have an exponential distribution if its probability density function(PDF) is given by,

Notation:

Moments

Variance

Coming back to our problem,

In this problem,

X = the usable lifetime of a randomly selected component.

(a) Here we need to find the proportion of these components that have a usable lifetime between 5.9 and 8.1 years.

X = the usable lifetime of a randomly selected component.

(b) Here we need to find the probability that that a randomly selected component will have a usable life more than 7.5 years.

X = the usable lifetime of a randomly selected component.

Hence the probability that that a randomly selected component will have a usable life more than 7.5 years is 0.2924

(c) The variance of X is given by,

X = the usable lifetime of a randomly selected component.

We know that,

Hence the variance of X is 37.21

orchestra answered 2 years ago
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